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Nondegenerate homoclinic tangency and hyperbolic sets
Nonlinear Analysis 52 (2003) 1521 – 1533
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Nondegenerate homoclinic tangency and hyperbolic sets Ming-Chia Li Department of Mathematics, National Changhua University of Education, Changhua 500, Taiwan Received 2 July 2001; accepted 6 March 2002
Abstract In this paper, we show that a nondegenerate homoclinic tangency on a surface is accumulated by a sequence of uniformly hyperbolic sets and in fact it is a boundary point of a nonuniformly hyperbolic set. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Nondegenerate homoclinic tangency; Uniform and nonuniformly hyperbolic sets; Cone 0eld; The H3enon map
1. De nitions and statements of theorems First of all, we give some basic de0nitions. Let M be a two-dimensional manifold with distance d. A point p is called a periodic saddle point of a di5eomorphism f on M if it is a periodic point with fn (p) = p for some n and the eigenvalues of Dfn (p), the derivative of fn at p, are two real numbers and with || ¡ 1 ¡ ||. Such a periodic saddle point is said to be dissipative if || ¡ 1. The stable and unstable manifolds of a periodic saddle point p are de0ned to be W s (p) = {z ∈ M : d(fn (z); fn (p)) → 0 as n → ∞} and W u (p) = {z ∈ M : d(fn (z); fn (p)) → 0 as n → − ∞};
E-mail address: [email protected] (M.-C. Li). 0362-546X/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 0 2 ) 0 0 2 7 1 - 7
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respectively. The stable manifold theorem [6] gives us that W s (p) and W u (p) are di5erentiable manifolds and transverse to each other at p, i.e., Tp W s (p) ⊕ Tp W u (p) = Tp M . Let p be a periodic saddle point of f. A point q ∈ M is called a homoclinic point for p if q ∈ W s (p) ∩ W u (p) \ {p}, i.e., q = p and limn→±∞ fn (q) = p. Such a point q is called a transverse homoclinic point provided the manifolds W s (p) and W u (p) have a transverse intersection at q, i.e., Tq W s (p) ⊕ Tq W u (p) = Tq M . If the intersection is not transverse, q is called a homoclinic tangency. In particular, we call q a nondegenerate homoclinic tangency if there are subarcs s ⊂ W s (p) and u ⊂ W u (p) such that q ∈ s ∩ u and there are local coordinates (x; y) near q with x(q) = y(q) = 0 and in which there are parameterizations {(x (t); y (t)) : |t| ¡ } of , =s; u, with ys (t)=0, yu (0) = yu (0) = 0, and yu (0) = 0. An invariant set for a di5eomorphism f on M has a uniformly hyperbolic structure if there exist subbundles Eu and Es of the tangent bundle TM and constants C ¿ 0 and ¿ 1 such that, for all z ∈ and n ¿ 0, Tz M = Eu (z) ⊕ Es (z); Df(z)Eu (z) = Eu (f(z)); Df(z)Es (z) = Es (f(z)); |Dfn (z)vu | ¿ Cn |vu |
for vu ∈ Eu (z);
and |Dfn (z)vs | 6 Cn |vs |
for vs ∈ Es (z):
If the above C and are functions of z with C(z) ¿ 0 and (z) ¿ 1 for all z ∈ , we say that has a nonuniformly hyperbolic structure. Refer to [14]. Some concepts and results about transverse homoclinic points have been presented since 1890 when Poincar3e [11] 0rst realized that transverse homoclinic orbits are accumulation points of other homoclinic points. Birkho5 [2] proved that every transverse homoclinic orbit of a two-dimensional di5eomorphism is accumulated by periodic orbits. Smale [15] generalized the result by showing that a transverse homoclinic orbit is contained in a hyperbolic set, the so-called “Smale’s horseshoe” now, in which the periodic points are dense. In this paper, we consider a nondegenerate homoclinic tangency on a surface and obtain results which describe local phenomena near the orbit of the tangency. First, we have that a nondegenerate homoclinic tangency is accumulated by a sequence of uniformly hyperbolic sets. Theorem 1. Let q be a nondegenerate homoclinic tangency for a dissipative periodic saddle point p of a di-eomorphism f on a two-dimensional manifold M . Then there is a positive integer N such that, for any integer n ¿ N , there exists a region Bn jn such that the invariant set n = ∩∞ j=−∞ f (Bn ) has a uniformly hyperbolic structure n for f . Moreover, q lies in closure( n¿N n ).
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Next, we go over to the 0rst return map which occurs in some region containing the homoclinic tangency, then get that the tangency is a boundary point of a nonuniformly hyperbolic set. Theorem 2. Under the same assumption as Theorem 1, there exists a region B such ∞ that if F is the /rst return map of f on B and B = j=−∞ F j (B) then ˜ = i i¿0 f (B ) has a nonuniformly hyperbolic structure for f. Furthermore, both p ˜ and q are on the boundary of . The above results are related to the H3enon map Hab (x; y)=(a−by −x2 ; x), especially for |b| not arbitrarily small. This map was 0rst introduced by H3enon [5]. Numerical studies indicate that there exist a nondegenerate homoclinic tangency and a “strange attractor” for b = −0:3 and a about 1.392. In [1], Benedicks and Carleson showed that there is a positive Lebesgue measure set E ⊂ (1; 2) such that if a ∈ E and b ¡ 0 with |b| small enough, then Hab has a “strange attractor”. In [16], Yang showed that, for b ¿ 0 small, before the 0rst homoclinic tangency occurs, the H3enon map has a hyperbolic structure on the nonwandering set and, right at the 0rst homoclinic tangency, the H3enon map is conjugate to the two-shift map with two orbits identi0ed. Both [1,16] are based on b very close to 0 and so the H3enon map is approximated by the one-dimensional system x → a − x2 . For |b| relatively large, e.g. b = −0:3 as above, many numerical indications need theoretical explanations. Refer to [10,13,14] for more discussions. Another interesting example which exhibits a homoclinic tangency is the Smale horseshoe composed with a translation downward so that the stable and unstable manifolds of a 0xed saddle point are indicated in Fig. 1. Readers may refer to [9,10] for
u
W (p)
s
W (p)
q
p
Fig. 1.
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more discussions on this example. See also [7,8,12] for recent results on destruction of uniformly hyperbolic invariant sets in parametrized families of di5eomorphisms through homoclinic tangencies occurring inside a horseshoe.
2. Preliminaries In order to avoid encumbering notional complexities, we shall present the discussion for an explicit form which retains the main features of the general cases. The model is due to Gavrilov and Shil3nikov [3,4]. First, without loss of generality, we assume that f is a di5eomorphism on R2 and p ∈ R2 is a dissipative 0xed saddle point such that Df(p) has two real eigenvalues and with 0 ¡ ¡ 1 ¡ ¡ −1 . Suppose the stable and unstable manifolds of p, W s (p) and W u (p), have a nondegenerate homoclinic tangency at a point q. From the theory of hyperbolicity, there is a neighborhood U of p and linearizing coordinates (x; y) on U such that p = (0; 0) and f(x; y) = Df(x; y) = (x; y) for all (x; y) ∈ U . Now, in the linearizing coordinates (x; y), W s (p) and W u (p) are segments of the x- and y-axis, respectively. The homoclinic tangency q lies on both W s (p) and W u (p). By looking along the orbit of q, we can assume that q = (x0 ; 0) is in U . Fix an integer k so large that f−k (q) = (0; y0 ) is in U . Denote q1 = f−k (q). See Fig. 2. Because the homoclinic tangency q is nondegenerate, we can assume further that fk is a quadratic mapping of the form fk (x; y) = (x0 + a(y − y0 ); −bx + c(y − y0 )2 ) in a neighborhood V of q1 , where a, b and c are positive constants. Here we drop the higher order terms in the form of fk (x; y). The reason for that is only for simplicity of
W u (p)
q1 =(0, y0)
W u (p)
W s (p)
p=(0,0)
q=(x0 ,0) Fig. 2.
M.-C. Li / Nonlinear Analysis 52 (2003) 1521 – 1533
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u
W (p) k
f (Bn) q
1
W u (p)
f n(Bn)
Bn q
p
W s (p)
Fig. 3.
discussion and this slight perturbation will not destroy the hyperbolic invariant sets n ˜ stated in the theorems, by the structural stability of hyperbolic invariant sets; and , refer to [14]. Next, we consider a strip near q in U so that fk is a quadratic map on some iteration of the strip. Take w ¿ 0 small, h ¿ 0 small, and m suOciently large so that the vertical strip y0 + h m m (x; y) ∈ U : (x0 − w) 6 x 6 (x0 + w) and 6 y 6 y0 + h near q1 = (0; y0 ) sits in V . For n = m + k, let Bn be a horizontal strip near q = (x0 ; 0) of the form y0 + h y0 + h Bn = (x; y) ∈ U : x0 − w 6 x 6 x0 + w and m+1 6 y 6 : m Since f is linear in the coordinate neighborhood U of p and fk is quadratic in the neighborhood V of q1 , we have that for (x; y) ∈ Bn , fn (x; y) = (x0 + a(m y − y0 ); −bm x + c(m y − y0 )2 ) and that fn (Bn ) = fk ◦ fm (Bn ) is a thin nonlinear box near q locally parallel to W u (p). See Fig. 3. We can assume that w; h and m are chosen so that fn (Bn ) stretches across Bn from top to bottom and back to the top again and that fn (Bn ) comes out the top of Bn and not the sides. In fact, the explicit conditions on w, h and m are −bm (x0 + w) + ch2 ¿ (y0 + h)=m and ah 6 w.
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a’
b’
d’
c’
d
a y0 +h λm
-
n
f (Bn)
y0 +h
Bn
λm+1 gn
c
b
2w Fig. 4.
Note that fn (Bn ) has the gap y0 + h gn = m+1 + bm (x0 − w); from the horizontal strip Bn . Furthermore, if z = (x; y) ∈ Bn ∩ fn (Bn ), then |x − x0 | ¿ a gn =c. See Fig. 4. 3. Proof of Theorem 1 We use the usual cone 0elds argument to prove the hyperbolicity structure; see [9, Theorem 3.1]. To do it, we follow the method of Robinson [13] to 0nd an unstable cone S(z) for z ∈ Bn ∩ fn (Bn ) which is invariant and expanded by Dfn (z). Let 0 ¡ 3; 4 ¡ 1 be constants. Since the gap gn = (y0 + h)=(m+1 ) + bm (x0 − w) and ¡ 1, there is a positive integer N large enough so that if n = m + k ¿ N , then √ 2 cgn 1 1 b m gn ¿ (2−3)m and (1 − 4) : ¿ √ a 4 2 cgn m From now on, we consider integers m and n so that n=m+k ¿ N . Let z=(x; y) ∈ Bn ∩ fn (Bn ) and z−1 = f−n (z) = (x−1 ; y−1 ). Note that the nonlinear map fn = fm+k carries the vertical line x˜ = x−1 into a parabola y˜ = −bm x−1 + (c=a2 )(x − x0 )2 . Thus the magnitude of the slope of any vector at z, which was originally vertical at z−1 in Bn , √ is |(2c=a2 )(x − x0 )| ¿ 2 cgn =a. Denote the slope by 2c 5(z) = 2 (x − x0 ) ; a then, for all z ∈ Bn ∩ fn (Bn ), 2c 2c 2√cgn 5(z) = 2 (x − x0 ) = (m y−1 − y0 ) ¿ : a a a The following lemma shows a particular relation between slopes 5(z) and 5(fn (z)).
M.-C. Li / Nonlinear Analysis 52 (2003) 1521 – 1533
Lemma 1. If z ∈ f−n (Bn ) ∩ Bn ∩ fn (Bn ), then 5(fn (z)) −
b m 1 ¿ 45(fn (z)): 45(z) a
Proof. Since (1 − 4)5(fn (z)) ¿ (1 − 4)
√ 2 cgn 1 b m ; ¿ √ a 4 2 cgn m
we have 5(fn (z)) −
1 b m ¿ 45(fn (z)): √ 4 2 cgn m
Therefore, 5(fn (z)) −
b m 1 b m 1 a ¿ 5(fn (z)) − √ 45(z) 4 2 cgn a a = 5(fn (z)) −
1 b m √ 4 2 cgn m
¿ 45(fn (z)): For z ∈ Bn ∩ fn (Bn ), de0ne the unstable cone at z by v2 S(z) = (v1 ; v2 ) ∈ Tz M : ¿ 45(z) : v1 For z = (x; y), we have 0 n Df (z) = −bm
am 2cm (m y − y0 )
:
Next, we show the invariance of the unstable cone under Dfn (z). Lemma 2. If z ∈ f−n (Bn ) ∩ Bn ∩ fn (Bn ), then Dfn (z)S(z) ⊂ S(fn (z)). Proof. Let z = (x; y), v = (v1 ; v2 ) ∈ S(z) and (v1 ; v2 ) = Dfn (z)v, then v2 −bm v1 + 2cm (m y − y0 )v2 2c(m y − y0 ) bm v1 = = − m v mv a a a v2 2 1 m 2c(m y − y0 ) bm v1 − = 5(fn (z)) − b v1 ¿ am v2 a a v2
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M.-C. Li / Nonlinear Analysis 52 (2003) 1521 – 1533
¿ 5(fn (z)) −
b m 1 45(z) a
¿ 45(fn (z)); where the last inequality follows from Lemma 1. Thus Dfn (z)v ∈ S(fn (z)). We use the norm (v1 ; v2 ) = max{|v1 |; |v2 |}. The expansion of the vectors in the unstable cone S(z) under Dfn (z) is shown as follows. Lemma 3. For any su2ciently large integer n, we have that Dfn (z)v ¿ v for all z ∈ f−n (Bn ) ∩ Bn ∩ fn (Bn ) and v ∈ S(z). Proof. Let z=(x; y) and v=(v1 ; v2 ). Since ¡ 1 and m y−y0 ¡ h, Dfn (z)v=am |v2 |. Hence √ √ Dfn (z)v am |v2 | m m 2 cgn ¿ |v | = a 45(z) ¿ a 4 ¿ 24 c3m ¿ 1; 2 v a 45(z)
where the last inequality holds by taking m and n = m + k suOciently large. Therefore, Dfn (z) expands the vectors in S(z) for all suOciently large n. Now we are ∞ready to prove Theorem 1. Let n = j=−∞ fjn (Bn ). From Lemma 2, we have that the cone S(z) is mapped inside the cone S(fn (z)). Hence there is an invariant bundle Eu (z) for z ∈ n . From Lemma 3, we have that the vectors v in Eu (z) are expanded by Dfn (z) for large n. Thus there are constants C ¿ 0 and ¿ 1 such that Dfn (z)v ¿ Cv for all z ∈ n and v ∈ Eu (z). Let S ∗ (z) be the complementary cone to S(z), then it is overQowing by Df−n (z) and so Df−n (z)S ∗ (z) ⊂ S ∗ (f−n (z)). Thus there is an invariant bundle Es (z). Since ¡ 1, det(Dfn (z)) ¡ 1. Because Dfn (z) is expanding on the invariant bundle Eu (z), it has to be contracting on Es (z). This shows that there exists a positive integer N such structure for fn . By the de0nitions that if n ¿ N then n has a uniformly hyperbolic of Bn and n , we have that q ∈ closure( n¿N n ). 4. Proof of Theorem 2 We shall 0rst consider a larger strip which contains q and have a part of its boundary on the local unstable manifold of p, then prove that there exists a nonuniformly hyperbolic set in this strip. Let k; w; h; N , and Bn be the same in Sections 2 and 3. De0ne y0 + h B = (x; y) ∈ U : x0 − w 6 x 6 x0 + w and 0 6 y 6 N −k : See Fig. 5. Because that points in B may return to B under di5erent iteration of f, we need to 0nd an unstable cone S(z), for z ∈ fn1 (Bn1 )∩Bn2 ∩f−n2 (Bn3 ) with arbitrary n1 ; n2 ; n3 ¿ N ,
M.-C. Li / Nonlinear Analysis 52 (2003) 1521 – 1533
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u
W (p) k
f (B) q1
Bn
B
p
W s (p)
q Fig. 5.
u
W (p) n
f 2( Bn 2) n
f 1( B n1)
Bn 1
W s (p)
B n2 q
gnn2 1
Fig. 6.
which is invariant and expanded by Dfn2 (z). The argument will be analogous to the proof of Theorem 1, but now the iterations of f and Df vary. Let ni = mi + k ¿ N for i = 1; 2; 3. Then the nonlinear box fn1 (Bn1 ) has the gap gnn12 =
y0 + h + bm1 (x0 − w); m2 +1
from the horizontal strip Bn2 , which satis0es gnn12 ¿ b(x0 − w): m1 If z = (x; y) ∈ fn1 (Bn1 ) ∩ Bn2 , then |x − x0 | ¿ a gnn12 =c. See Fig. 6. m2 gnn12 ¿
y0 + h
and
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Let 0 ¡ 7 ¡ 1 be a constant. Replacing N by a larger integer if necessary, we can assume that b 2 N −k 1 (1 − 7) bc(x0 − w) ¿ : a 7 2 c(y0 + h)= N −k For z = (x; y) ∈ fn1 (Bn1 ) ∩ Bn2 , de0ne 2c 5(z) = 2 (x − x0 ) a and
S(z) = (v1 ; v2 ) ∈ Tz M
v2 : ¿ 75(z) : v1
We have a relation between 5(z) and 5(fn2 (z)). Lemma 4. If z ∈ fn1 (Bn1 ) ∩ Bn2 ∩ f−n2 (Bn3 ), then b n2 1 5(fn2 (z)) − ¿ 75(fn2 (z)): a 75(z) Proof. Since
¿
1 7
=
1 7
cgnn23 2 ¿ (1 − 7) bc(x0 − w) m 2 a b b N −k n2 −k 1 ¿ N −k 7 2 c(y0 + h)= n2 −k 2 c(y0 + h)= b m2 ; 2 c(y0 + h)= m2
5(fn2 (z)) 2 (1 − 7) √ m2 ¿ (1 − 7) a
we get b 5(fn2 (z)) 1 √ m − 7 2 c(y0 + h)= 2
m2 5(fn2 (z)) √ m : ¿ 7 m2 2
Therefore,
b m2 1 a 75(z) b m2 ¿ 5(fn2 (z)) − a √ 5(fn2 (z)) 1 √ m = m2 − 2 7
5(fn2 (z)) −
a 1 7 2 cgnn12
√ m
b 2 n 2 cgn12 m2
m2 √ m 5(fn2 (z)) 1 b √ m = 2 − 2 7 2 cgnn12 m2 m2
M.-C. Li / Nonlinear Analysis 52 (2003) 1521 – 1533
√
= m2
b 5(fn2 (z)) 1 √ m − 2 7 2 c(y0 + h)=
m2 m2
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√ 5(fn2 (z)) ¿ m2 7 √ m2 ¿ 75(fn2 (z)): The following lemma shows that the unstable cone S(z) is invariant under Dfn2 (z). Lemma 5. If z ∈ fn1 (Bn1 ) ∩ Bn2 ∩ f−n2 (Bn3 ), then Dfn2 (z)S(z) ⊂ S(fn2 (z)). Proof. Let z = (x; y), v = (v1 ; v2 ) ∈ S(z) and (v ; v2 ) = Dfn2 (z)v, then v2 −bm2 v1 + 2cm2 (m2 y − y0 )v2 = v am2 v2 1
2c m bm2 v1 2 = ( y − y0 ) − m2 a v2 a m b 2 v1 ¿ 5(fn2 (z)) − m2 a v2 b m2 1 ¿ 5(fn2 (z)) − a 75(z) ¿ 75(fn2 (z));
where the last inequality follows from Lemma 4. Thus Dfn2 (z)v ∈ S(fn2 (z)). The vectors in the unstable cone S(z) are expanded by Dfn2 (z). Lemma 6. For any su2ciently large integer n2 , we have that Dfn2 (z)v ¿ v for all z ∈ fn1 (Bn1 ) ∩ Bn2 ∩ f−n2 (Bn3 ) and v ∈ S(z). Proof. Let z = (x; y) and v = (v1 ; v2 ), then Dfn2 (z)v = am2 |v2 |. Hence 2 cgnn12 am2 |v2 | Dfn2 (z)v ¿ |v | ¿ am2 7 = 27 c2m2 gnn12 2 v a 75(z)
¿ 27
cm2 −1 (y0 + h)
¿ 1; where the last inequality holds by taking m2 and n2 = m2 + k suOciently large. Thus Dfn2 (z) expands the vectors in S(z). Now we prove Theorem 2.
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M.-C. Li / Nonlinear Analysis 52 (2003) 1521 – 1533
∞ Let F be the 0rst return map of f on B and let B = j=−∞ F j (B) and z ∈ B . There are least positive integers n = n(z) and n = n (z) such that F(z) = fn (z) ∈ B and F −1 (z) = f−n (z) ∈ B. From Lemma 5, the cone S(z) is mapped inside the cone S(F(z)) = S(fn (z)). Hence there is an invariant bundle Eu (z) for the 0rst return map F. From Lemma 6, the vectors v in Eu (z) are expanded by DF(z) = Dfn (z). Since n=n(z) depends on z, there are functions C(z) ¿ 0 and (z) ¿ 1 such that DF(z)v= Dfn (z)v ¿ C(z)(z)v for all v ∈ Eu (z). Therefore, for all j ¿ 0, DF j (z)v ¿ C j (z)j (z)v, where C j (z) = C(F j−1 (z))C(F j−2 (z)) · · · C(z) and j (z) = (F j−1 (z)) (F j−2 (z)) · · · (z). Let S ∗ (z) be the complimentary cone to S(z), then it is overQowing by DF −1 (z) = Df−n (z) and so DF −1 (z) = Df−n (z)S ∗ (z) ⊂ S ∗ (f−n (z)) = S ∗ (F −1 (z)). Thus there is an invariant bundle Es (z) for the 0rst return map F. Since ¡ 1, det(DF(z)) ¡ 1. Because DF(z) is expanding on the invariant bundle Eu (z), it has to be contracting s on E (z). This shows that B has a nonuniformly hyperbolic structure for F and so ˜ = i¿0 fi (B ) is a nonuniformly hyperbolic structure for f. It is obvious that both ˜ p and q are boundary points of . Acknowledgement The author is indebted to Professor Clark Robinson of Northwestern University for proposing this theme and invaluable conversations.
References [1] M. Benedicks, L. Carleson, The dynamics of the H3enon map, Ann. of Math. 133 (1991) 73–169. [2] G.D. Birkho5, Nouvelles recherches sur les systRems dynamiques, Mem. Pont. Acad. Sci. Novi. Lyncaei 1 (1935) 85–216. [3] N.K. Gavrilov, L.P. Shil3nikov, On three-dimensional dynamical systems close to systems with structurally unstable homoclinic curve I, Math. USSR Sbornik 17 (1972) 467–485. [4] N.K. Gavrilov, L.P. Shil3nikov, On three-dimensional dynamical systems close to systems with structurally unstable homoclinic curve II, Math. USSR Sbornik 19 (1973) 139–156. [5] M. H3enon, A two-dimensional mapping with a strange attractor, Comm. Math. Phys. 50 (1976) 69–77. [6] M. Hirsch, C. Pugh, Stable manifolds and hyperbolic sets, Global Analysis, Proceedings of the Symposium in Pure Mathematics, Vol. 14, American Mathematical Society, Providence, RI, 1970, pp. 133–163. [7] S. Kiriki, Hyperbolicity of the special one-parameter family with a 0rst homoclinic tangency, Dyn. Stability Systems 11 (1996) 3–18. [8] S. Kiriki, The Palis and Takens’ problem on the 0rst homoclinic tangency inside the horseshoe, Internat. J. Bifurc. Chaos 6 (1996) 737–744. [9] S. Newhouse, J. Palis, Bifurcations of Morse–Smale dynamical systems, in: M. Peixoto (Ed.), Dynamical Systems, Academic Press, New York, 1973, pp. 303–366. [10] J. Palis, F. Takens, Hyperbolic and Sensitive Chaotic Dynamics at Homoclinic Bifurcations, Cambridge Studies in Advanced Mathematics, Vol. 35, Cambridge University Press, Cambridge, Great Britain, 1993. [11] H. Poincar3e, Sur les e3 quations de la dynamique et le problReme de trois corps, Acta Math. 13 (1890) 1–270.
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[12] I.L. Rios, Unfolding homoclinic tangencies inside horseshoes: hyperbolicity, fractal dimensions and persistent tangencies, Nonlinearity 14 (2001) 431–462. [13] C. Robinson, Bifurcation to in0nity many sinks, Comm. Math. Phys. 90 (1983) 433–459. [14] C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, 2nd Edition, CRC Press, Boca Raton, FL, 1999. [15] S. Smale, Di5eomorphisms with many periodic points, Di5erential and Combinatorial Topology, Princeton University Press, Princeton, NJ, 1965, pp. 63–80. [16] Z. Yang, Henon-like maps before or at the 0rst tangency, Ph.D. Thesis, University of North Carolina, Chapel Hill, 1995.
www.elsevier.com/locate/na
Nondegenerate homoclinic tangency and hyperbolic sets Ming-Chia Li Department of Mathematics, National Changhua University of Education, Changhua 500, Taiwan Received 2 July 2001; accepted 6 March 2002
Abstract In this paper, we show that a nondegenerate homoclinic tangency on a surface is accumulated by a sequence of uniformly hyperbolic sets and in fact it is a boundary point of a nonuniformly hyperbolic set. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Nondegenerate homoclinic tangency; Uniform and nonuniformly hyperbolic sets; Cone 0eld; The H3enon map
1. De nitions and statements of theorems First of all, we give some basic de0nitions. Let M be a two-dimensional manifold with distance d. A point p is called a periodic saddle point of a di5eomorphism f on M if it is a periodic point with fn (p) = p for some n and the eigenvalues of Dfn (p), the derivative of fn at p, are two real numbers and with || ¡ 1 ¡ ||. Such a periodic saddle point is said to be dissipative if || ¡ 1. The stable and unstable manifolds of a periodic saddle point p are de0ned to be W s (p) = {z ∈ M : d(fn (z); fn (p)) → 0 as n → ∞} and W u (p) = {z ∈ M : d(fn (z); fn (p)) → 0 as n → − ∞};
E-mail address: [email protected] (M.-C. Li). 0362-546X/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 0 2 ) 0 0 2 7 1 - 7
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respectively. The stable manifold theorem [6] gives us that W s (p) and W u (p) are di5erentiable manifolds and transverse to each other at p, i.e., Tp W s (p) ⊕ Tp W u (p) = Tp M . Let p be a periodic saddle point of f. A point q ∈ M is called a homoclinic point for p if q ∈ W s (p) ∩ W u (p) \ {p}, i.e., q = p and limn→±∞ fn (q) = p. Such a point q is called a transverse homoclinic point provided the manifolds W s (p) and W u (p) have a transverse intersection at q, i.e., Tq W s (p) ⊕ Tq W u (p) = Tq M . If the intersection is not transverse, q is called a homoclinic tangency. In particular, we call q a nondegenerate homoclinic tangency if there are subarcs s ⊂ W s (p) and u ⊂ W u (p) such that q ∈ s ∩ u and there are local coordinates (x; y) near q with x(q) = y(q) = 0 and in which there are parameterizations {(x (t); y (t)) : |t| ¡ } of , =s; u, with ys (t)=0, yu (0) = yu (0) = 0, and yu (0) = 0. An invariant set for a di5eomorphism f on M has a uniformly hyperbolic structure if there exist subbundles Eu and Es of the tangent bundle TM and constants C ¿ 0 and ¿ 1 such that, for all z ∈ and n ¿ 0, Tz M = Eu (z) ⊕ Es (z); Df(z)Eu (z) = Eu (f(z)); Df(z)Es (z) = Es (f(z)); |Dfn (z)vu | ¿ Cn |vu |
for vu ∈ Eu (z);
and |Dfn (z)vs | 6 Cn |vs |
for vs ∈ Es (z):
If the above C and are functions of z with C(z) ¿ 0 and (z) ¿ 1 for all z ∈ , we say that has a nonuniformly hyperbolic structure. Refer to [14]. Some concepts and results about transverse homoclinic points have been presented since 1890 when Poincar3e [11] 0rst realized that transverse homoclinic orbits are accumulation points of other homoclinic points. Birkho5 [2] proved that every transverse homoclinic orbit of a two-dimensional di5eomorphism is accumulated by periodic orbits. Smale [15] generalized the result by showing that a transverse homoclinic orbit is contained in a hyperbolic set, the so-called “Smale’s horseshoe” now, in which the periodic points are dense. In this paper, we consider a nondegenerate homoclinic tangency on a surface and obtain results which describe local phenomena near the orbit of the tangency. First, we have that a nondegenerate homoclinic tangency is accumulated by a sequence of uniformly hyperbolic sets. Theorem 1. Let q be a nondegenerate homoclinic tangency for a dissipative periodic saddle point p of a di-eomorphism f on a two-dimensional manifold M . Then there is a positive integer N such that, for any integer n ¿ N , there exists a region Bn jn such that the invariant set n = ∩∞ j=−∞ f (Bn ) has a uniformly hyperbolic structure n for f . Moreover, q lies in closure( n¿N n ).
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Next, we go over to the 0rst return map which occurs in some region containing the homoclinic tangency, then get that the tangency is a boundary point of a nonuniformly hyperbolic set. Theorem 2. Under the same assumption as Theorem 1, there exists a region B such ∞ that if F is the /rst return map of f on B and B = j=−∞ F j (B) then ˜ = i i¿0 f (B ) has a nonuniformly hyperbolic structure for f. Furthermore, both p ˜ and q are on the boundary of . The above results are related to the H3enon map Hab (x; y)=(a−by −x2 ; x), especially for |b| not arbitrarily small. This map was 0rst introduced by H3enon [5]. Numerical studies indicate that there exist a nondegenerate homoclinic tangency and a “strange attractor” for b = −0:3 and a about 1.392. In [1], Benedicks and Carleson showed that there is a positive Lebesgue measure set E ⊂ (1; 2) such that if a ∈ E and b ¡ 0 with |b| small enough, then Hab has a “strange attractor”. In [16], Yang showed that, for b ¿ 0 small, before the 0rst homoclinic tangency occurs, the H3enon map has a hyperbolic structure on the nonwandering set and, right at the 0rst homoclinic tangency, the H3enon map is conjugate to the two-shift map with two orbits identi0ed. Both [1,16] are based on b very close to 0 and so the H3enon map is approximated by the one-dimensional system x → a − x2 . For |b| relatively large, e.g. b = −0:3 as above, many numerical indications need theoretical explanations. Refer to [10,13,14] for more discussions. Another interesting example which exhibits a homoclinic tangency is the Smale horseshoe composed with a translation downward so that the stable and unstable manifolds of a 0xed saddle point are indicated in Fig. 1. Readers may refer to [9,10] for
u
W (p)
s
W (p)
q
p
Fig. 1.
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more discussions on this example. See also [7,8,12] for recent results on destruction of uniformly hyperbolic invariant sets in parametrized families of di5eomorphisms through homoclinic tangencies occurring inside a horseshoe.
2. Preliminaries In order to avoid encumbering notional complexities, we shall present the discussion for an explicit form which retains the main features of the general cases. The model is due to Gavrilov and Shil3nikov [3,4]. First, without loss of generality, we assume that f is a di5eomorphism on R2 and p ∈ R2 is a dissipative 0xed saddle point such that Df(p) has two real eigenvalues and with 0 ¡ ¡ 1 ¡ ¡ −1 . Suppose the stable and unstable manifolds of p, W s (p) and W u (p), have a nondegenerate homoclinic tangency at a point q. From the theory of hyperbolicity, there is a neighborhood U of p and linearizing coordinates (x; y) on U such that p = (0; 0) and f(x; y) = Df(x; y) = (x; y) for all (x; y) ∈ U . Now, in the linearizing coordinates (x; y), W s (p) and W u (p) are segments of the x- and y-axis, respectively. The homoclinic tangency q lies on both W s (p) and W u (p). By looking along the orbit of q, we can assume that q = (x0 ; 0) is in U . Fix an integer k so large that f−k (q) = (0; y0 ) is in U . Denote q1 = f−k (q). See Fig. 2. Because the homoclinic tangency q is nondegenerate, we can assume further that fk is a quadratic mapping of the form fk (x; y) = (x0 + a(y − y0 ); −bx + c(y − y0 )2 ) in a neighborhood V of q1 , where a, b and c are positive constants. Here we drop the higher order terms in the form of fk (x; y). The reason for that is only for simplicity of
W u (p)
q1 =(0, y0)
W u (p)
W s (p)
p=(0,0)
q=(x0 ,0) Fig. 2.
M.-C. Li / Nonlinear Analysis 52 (2003) 1521 – 1533
1525
u
W (p) k
f (Bn) q
1
W u (p)
f n(Bn)
Bn q
p
W s (p)
Fig. 3.
discussion and this slight perturbation will not destroy the hyperbolic invariant sets n ˜ stated in the theorems, by the structural stability of hyperbolic invariant sets; and , refer to [14]. Next, we consider a strip near q in U so that fk is a quadratic map on some iteration of the strip. Take w ¿ 0 small, h ¿ 0 small, and m suOciently large so that the vertical strip y0 + h m m (x; y) ∈ U : (x0 − w) 6 x 6 (x0 + w) and 6 y 6 y0 + h near q1 = (0; y0 ) sits in V . For n = m + k, let Bn be a horizontal strip near q = (x0 ; 0) of the form y0 + h y0 + h Bn = (x; y) ∈ U : x0 − w 6 x 6 x0 + w and m+1 6 y 6 : m Since f is linear in the coordinate neighborhood U of p and fk is quadratic in the neighborhood V of q1 , we have that for (x; y) ∈ Bn , fn (x; y) = (x0 + a(m y − y0 ); −bm x + c(m y − y0 )2 ) and that fn (Bn ) = fk ◦ fm (Bn ) is a thin nonlinear box near q locally parallel to W u (p). See Fig. 3. We can assume that w; h and m are chosen so that fn (Bn ) stretches across Bn from top to bottom and back to the top again and that fn (Bn ) comes out the top of Bn and not the sides. In fact, the explicit conditions on w, h and m are −bm (x0 + w) + ch2 ¿ (y0 + h)=m and ah 6 w.
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a’
b’
d’
c’
d
a y0 +h λm
-
n
f (Bn)
y0 +h
Bn
λm+1 gn
c
b
2w Fig. 4.
Note that fn (Bn ) has the gap y0 + h gn = m+1 + bm (x0 − w); from the horizontal strip Bn . Furthermore, if z = (x; y) ∈ Bn ∩ fn (Bn ), then |x − x0 | ¿ a gn =c. See Fig. 4. 3. Proof of Theorem 1 We use the usual cone 0elds argument to prove the hyperbolicity structure; see [9, Theorem 3.1]. To do it, we follow the method of Robinson [13] to 0nd an unstable cone S(z) for z ∈ Bn ∩ fn (Bn ) which is invariant and expanded by Dfn (z). Let 0 ¡ 3; 4 ¡ 1 be constants. Since the gap gn = (y0 + h)=(m+1 ) + bm (x0 − w) and ¡ 1, there is a positive integer N large enough so that if n = m + k ¿ N , then √ 2 cgn 1 1 b m gn ¿ (2−3)m and (1 − 4) : ¿ √ a 4 2 cgn m From now on, we consider integers m and n so that n=m+k ¿ N . Let z=(x; y) ∈ Bn ∩ fn (Bn ) and z−1 = f−n (z) = (x−1 ; y−1 ). Note that the nonlinear map fn = fm+k carries the vertical line x˜ = x−1 into a parabola y˜ = −bm x−1 + (c=a2 )(x − x0 )2 . Thus the magnitude of the slope of any vector at z, which was originally vertical at z−1 in Bn , √ is |(2c=a2 )(x − x0 )| ¿ 2 cgn =a. Denote the slope by 2c 5(z) = 2 (x − x0 ) ; a then, for all z ∈ Bn ∩ fn (Bn ), 2c 2c 2√cgn 5(z) = 2 (x − x0 ) = (m y−1 − y0 ) ¿ : a a a The following lemma shows a particular relation between slopes 5(z) and 5(fn (z)).
M.-C. Li / Nonlinear Analysis 52 (2003) 1521 – 1533
Lemma 1. If z ∈ f−n (Bn ) ∩ Bn ∩ fn (Bn ), then 5(fn (z)) −
b m 1 ¿ 45(fn (z)): 45(z) a
Proof. Since (1 − 4)5(fn (z)) ¿ (1 − 4)
√ 2 cgn 1 b m ; ¿ √ a 4 2 cgn m
we have 5(fn (z)) −
1 b m ¿ 45(fn (z)): √ 4 2 cgn m
Therefore, 5(fn (z)) −
b m 1 b m 1 a ¿ 5(fn (z)) − √ 45(z) 4 2 cgn a a = 5(fn (z)) −
1 b m √ 4 2 cgn m
¿ 45(fn (z)): For z ∈ Bn ∩ fn (Bn ), de0ne the unstable cone at z by v2 S(z) = (v1 ; v2 ) ∈ Tz M : ¿ 45(z) : v1 For z = (x; y), we have 0 n Df (z) = −bm
am 2cm (m y − y0 )
:
Next, we show the invariance of the unstable cone under Dfn (z). Lemma 2. If z ∈ f−n (Bn ) ∩ Bn ∩ fn (Bn ), then Dfn (z)S(z) ⊂ S(fn (z)). Proof. Let z = (x; y), v = (v1 ; v2 ) ∈ S(z) and (v1 ; v2 ) = Dfn (z)v, then v2 −bm v1 + 2cm (m y − y0 )v2 2c(m y − y0 ) bm v1 = = − m v mv a a a v2 2 1 m 2c(m y − y0 ) bm v1 − = 5(fn (z)) − b v1 ¿ am v2 a a v2
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M.-C. Li / Nonlinear Analysis 52 (2003) 1521 – 1533
¿ 5(fn (z)) −
b m 1 45(z) a
¿ 45(fn (z)); where the last inequality follows from Lemma 1. Thus Dfn (z)v ∈ S(fn (z)). We use the norm (v1 ; v2 ) = max{|v1 |; |v2 |}. The expansion of the vectors in the unstable cone S(z) under Dfn (z) is shown as follows. Lemma 3. For any su2ciently large integer n, we have that Dfn (z)v ¿ v for all z ∈ f−n (Bn ) ∩ Bn ∩ fn (Bn ) and v ∈ S(z). Proof. Let z=(x; y) and v=(v1 ; v2 ). Since ¡ 1 and m y−y0 ¡ h, Dfn (z)v=am |v2 |. Hence √ √ Dfn (z)v am |v2 | m m 2 cgn ¿ |v | = a 45(z) ¿ a 4 ¿ 24 c3m ¿ 1; 2 v a 45(z)
where the last inequality holds by taking m and n = m + k suOciently large. Therefore, Dfn (z) expands the vectors in S(z) for all suOciently large n. Now we are ∞ready to prove Theorem 1. Let n = j=−∞ fjn (Bn ). From Lemma 2, we have that the cone S(z) is mapped inside the cone S(fn (z)). Hence there is an invariant bundle Eu (z) for z ∈ n . From Lemma 3, we have that the vectors v in Eu (z) are expanded by Dfn (z) for large n. Thus there are constants C ¿ 0 and ¿ 1 such that Dfn (z)v ¿ Cv for all z ∈ n and v ∈ Eu (z). Let S ∗ (z) be the complementary cone to S(z), then it is overQowing by Df−n (z) and so Df−n (z)S ∗ (z) ⊂ S ∗ (f−n (z)). Thus there is an invariant bundle Es (z). Since ¡ 1, det(Dfn (z)) ¡ 1. Because Dfn (z) is expanding on the invariant bundle Eu (z), it has to be contracting on Es (z). This shows that there exists a positive integer N such structure for fn . By the de0nitions that if n ¿ N then n has a uniformly hyperbolic of Bn and n , we have that q ∈ closure( n¿N n ). 4. Proof of Theorem 2 We shall 0rst consider a larger strip which contains q and have a part of its boundary on the local unstable manifold of p, then prove that there exists a nonuniformly hyperbolic set in this strip. Let k; w; h; N , and Bn be the same in Sections 2 and 3. De0ne y0 + h B = (x; y) ∈ U : x0 − w 6 x 6 x0 + w and 0 6 y 6 N −k : See Fig. 5. Because that points in B may return to B under di5erent iteration of f, we need to 0nd an unstable cone S(z), for z ∈ fn1 (Bn1 )∩Bn2 ∩f−n2 (Bn3 ) with arbitrary n1 ; n2 ; n3 ¿ N ,
M.-C. Li / Nonlinear Analysis 52 (2003) 1521 – 1533
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u
W (p) k
f (B) q1
Bn
B
p
W s (p)
q Fig. 5.
u
W (p) n
f 2( Bn 2) n
f 1( B n1)
Bn 1
W s (p)
B n2 q
gnn2 1
Fig. 6.
which is invariant and expanded by Dfn2 (z). The argument will be analogous to the proof of Theorem 1, but now the iterations of f and Df vary. Let ni = mi + k ¿ N for i = 1; 2; 3. Then the nonlinear box fn1 (Bn1 ) has the gap gnn12 =
y0 + h + bm1 (x0 − w); m2 +1
from the horizontal strip Bn2 , which satis0es gnn12 ¿ b(x0 − w): m1 If z = (x; y) ∈ fn1 (Bn1 ) ∩ Bn2 , then |x − x0 | ¿ a gnn12 =c. See Fig. 6. m2 gnn12 ¿
y0 + h
and
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Let 0 ¡ 7 ¡ 1 be a constant. Replacing N by a larger integer if necessary, we can assume that b 2 N −k 1 (1 − 7) bc(x0 − w) ¿ : a 7 2 c(y0 + h)= N −k For z = (x; y) ∈ fn1 (Bn1 ) ∩ Bn2 , de0ne 2c 5(z) = 2 (x − x0 ) a and
S(z) = (v1 ; v2 ) ∈ Tz M
v2 : ¿ 75(z) : v1
We have a relation between 5(z) and 5(fn2 (z)). Lemma 4. If z ∈ fn1 (Bn1 ) ∩ Bn2 ∩ f−n2 (Bn3 ), then b n2 1 5(fn2 (z)) − ¿ 75(fn2 (z)): a 75(z) Proof. Since
¿
1 7
=
1 7
cgnn23 2 ¿ (1 − 7) bc(x0 − w) m 2 a b b N −k n2 −k 1 ¿ N −k 7 2 c(y0 + h)= n2 −k 2 c(y0 + h)= b m2 ; 2 c(y0 + h)= m2
5(fn2 (z)) 2 (1 − 7) √ m2 ¿ (1 − 7) a
we get b 5(fn2 (z)) 1 √ m − 7 2 c(y0 + h)= 2
m2 5(fn2 (z)) √ m : ¿ 7 m2 2
Therefore,
b m2 1 a 75(z) b m2 ¿ 5(fn2 (z)) − a √ 5(fn2 (z)) 1 √ m = m2 − 2 7
5(fn2 (z)) −
a 1 7 2 cgnn12
√ m
b 2 n 2 cgn12 m2
m2 √ m 5(fn2 (z)) 1 b √ m = 2 − 2 7 2 cgnn12 m2 m2
M.-C. Li / Nonlinear Analysis 52 (2003) 1521 – 1533
√
= m2
b 5(fn2 (z)) 1 √ m − 2 7 2 c(y0 + h)=
m2 m2
1531
√ 5(fn2 (z)) ¿ m2 7 √ m2 ¿ 75(fn2 (z)): The following lemma shows that the unstable cone S(z) is invariant under Dfn2 (z). Lemma 5. If z ∈ fn1 (Bn1 ) ∩ Bn2 ∩ f−n2 (Bn3 ), then Dfn2 (z)S(z) ⊂ S(fn2 (z)). Proof. Let z = (x; y), v = (v1 ; v2 ) ∈ S(z) and (v ; v2 ) = Dfn2 (z)v, then v2 −bm2 v1 + 2cm2 (m2 y − y0 )v2 = v am2 v2 1
2c m bm2 v1 2 = ( y − y0 ) − m2 a v2 a m b 2 v1 ¿ 5(fn2 (z)) − m2 a v2 b m2 1 ¿ 5(fn2 (z)) − a 75(z) ¿ 75(fn2 (z));
where the last inequality follows from Lemma 4. Thus Dfn2 (z)v ∈ S(fn2 (z)). The vectors in the unstable cone S(z) are expanded by Dfn2 (z). Lemma 6. For any su2ciently large integer n2 , we have that Dfn2 (z)v ¿ v for all z ∈ fn1 (Bn1 ) ∩ Bn2 ∩ f−n2 (Bn3 ) and v ∈ S(z). Proof. Let z = (x; y) and v = (v1 ; v2 ), then Dfn2 (z)v = am2 |v2 |. Hence 2 cgnn12 am2 |v2 | Dfn2 (z)v ¿ |v | ¿ am2 7 = 27 c2m2 gnn12 2 v a 75(z)
¿ 27
cm2 −1 (y0 + h)
¿ 1; where the last inequality holds by taking m2 and n2 = m2 + k suOciently large. Thus Dfn2 (z) expands the vectors in S(z). Now we prove Theorem 2.
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∞ Let F be the 0rst return map of f on B and let B = j=−∞ F j (B) and z ∈ B . There are least positive integers n = n(z) and n = n (z) such that F(z) = fn (z) ∈ B and F −1 (z) = f−n (z) ∈ B. From Lemma 5, the cone S(z) is mapped inside the cone S(F(z)) = S(fn (z)). Hence there is an invariant bundle Eu (z) for the 0rst return map F. From Lemma 6, the vectors v in Eu (z) are expanded by DF(z) = Dfn (z). Since n=n(z) depends on z, there are functions C(z) ¿ 0 and (z) ¿ 1 such that DF(z)v= Dfn (z)v ¿ C(z)(z)v for all v ∈ Eu (z). Therefore, for all j ¿ 0, DF j (z)v ¿ C j (z)j (z)v, where C j (z) = C(F j−1 (z))C(F j−2 (z)) · · · C(z) and j (z) = (F j−1 (z)) (F j−2 (z)) · · · (z). Let S ∗ (z) be the complimentary cone to S(z), then it is overQowing by DF −1 (z) = Df−n (z) and so DF −1 (z) = Df−n (z)S ∗ (z) ⊂ S ∗ (f−n (z)) = S ∗ (F −1 (z)). Thus there is an invariant bundle Es (z) for the 0rst return map F. Since ¡ 1, det(DF(z)) ¡ 1. Because DF(z) is expanding on the invariant bundle Eu (z), it has to be contracting s on E (z). This shows that B has a nonuniformly hyperbolic structure for F and so ˜ = i¿0 fi (B ) is a nonuniformly hyperbolic structure for f. It is obvious that both ˜ p and q are boundary points of . Acknowledgement The author is indebted to Professor Clark Robinson of Northwestern University for proposing this theme and invaluable conversations.
References [1] M. Benedicks, L. Carleson, The dynamics of the H3enon map, Ann. of Math. 133 (1991) 73–169. [2] G.D. Birkho5, Nouvelles recherches sur les systRems dynamiques, Mem. Pont. Acad. Sci. Novi. Lyncaei 1 (1935) 85–216. [3] N.K. Gavrilov, L.P. Shil3nikov, On three-dimensional dynamical systems close to systems with structurally unstable homoclinic curve I, Math. USSR Sbornik 17 (1972) 467–485. [4] N.K. Gavrilov, L.P. Shil3nikov, On three-dimensional dynamical systems close to systems with structurally unstable homoclinic curve II, Math. USSR Sbornik 19 (1973) 139–156. [5] M. H3enon, A two-dimensional mapping with a strange attractor, Comm. Math. Phys. 50 (1976) 69–77. [6] M. Hirsch, C. Pugh, Stable manifolds and hyperbolic sets, Global Analysis, Proceedings of the Symposium in Pure Mathematics, Vol. 14, American Mathematical Society, Providence, RI, 1970, pp. 133–163. [7] S. Kiriki, Hyperbolicity of the special one-parameter family with a 0rst homoclinic tangency, Dyn. Stability Systems 11 (1996) 3–18. [8] S. Kiriki, The Palis and Takens’ problem on the 0rst homoclinic tangency inside the horseshoe, Internat. J. Bifurc. Chaos 6 (1996) 737–744. [9] S. Newhouse, J. Palis, Bifurcations of Morse–Smale dynamical systems, in: M. Peixoto (Ed.), Dynamical Systems, Academic Press, New York, 1973, pp. 303–366. [10] J. Palis, F. Takens, Hyperbolic and Sensitive Chaotic Dynamics at Homoclinic Bifurcations, Cambridge Studies in Advanced Mathematics, Vol. 35, Cambridge University Press, Cambridge, Great Britain, 1993. [11] H. Poincar3e, Sur les e3 quations de la dynamique et le problReme de trois corps, Acta Math. 13 (1890) 1–270.
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[12] I.L. Rios, Unfolding homoclinic tangencies inside horseshoes: hyperbolicity, fractal dimensions and persistent tangencies, Nonlinearity 14 (2001) 431–462. [13] C. Robinson, Bifurcation to in0nity many sinks, Comm. Math. Phys. 90 (1983) 433–459. [14] C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, 2nd Edition, CRC Press, Boca Raton, FL, 1999. [15] S. Smale, Di5eomorphisms with many periodic points, Di5erential and Combinatorial Topology, Princeton University Press, Princeton, NJ, 1965, pp. 63–80. [16] Z. Yang, Henon-like maps before or at the 0rst tangency, Ph.D. Thesis, University of North Carolina, Chapel Hill, 1995.